The Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and differences of subharmonic functions

4 Jul 2021, 10:00


Bulat Khabibullin (Bashkir State University)


Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the maximum function of its modulus $M(r,f)$ on circles centered at zero of radius $r$. A number of classical, well-known and widely used results allow us to estimate from above the integrals of the positive part of the logarithm ln M(t,f) over subsets of $E\subset [0,r]$ via the Nevanlinna characteristic T(r,f) and the linear Lebesgue measure of the set $E$. We give much more general estimates for the Lebesgue-Stieltjes integrals of $ln^+M(t,f)$ over the increasing integration function of $m$. Our results are established immediately for the differences of subharmonic functions on closed circles centered at zero, i.e., $ \delta $ -subharmonic functions, but they are new for meromorphic functions on $\mathbb C$ and contain all the previous results on this topic as a very special and extreme case. The only condition in our main theorem is the Dini condition for the modulus of continuity of the integration function $m$. This condition is, in a sense, necessary. Thus, our results, to a certain extent, complete study upper estimates of the integrals of the radial maximum growth characteristics of arbitrary meromorphic and $\delta$-subharmonic functions through the Nevanlinna characteristic with its versions and through quantities associated with the integration function $m$ such as the Hausdorff $h$-measure or $h$-content, and the $d$-dimensional Hausdorff measure of the support of nonconstancy for $m$.

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Bulat Khabibullin (Bashkir State University)

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